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8.4: Fractional Exponents

Difficulty Level: Basic Created by: CK-12
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Suppose you want to find the cube root of a number, but your calculator only allows you to find square roots. However, it does allow you to find a number raised to any power, including non-integer powers. What would you do? Is there a way that you could rewrite the cube root of a number with a fractional exponent so that you could use your calculator to find the answer? In this Concept, you'll learn how to rewrite roots as fractional exponents so that you won't need to rely on your calculator's root button.


The next objective is to be able to use fractions as exponents in an expression.

Roots as Fractional Exponents: \begin{align*}\sqrt[m]{a^n}=a^{\frac{n}{m}}\end{align*}anm=anm.

Example A

\begin{align*}\sqrt{a}=a^{\frac{1}{2}}, \sqrt[3]{a}=a^{\frac{1}{3}}, \sqrt[5]{a^2}=(a^2)^{\frac{1}{5}}=a^{\frac{2}{5}}\end{align*}a=a12,a3=a13,a25=(a2)15=a25

Example B

Simplify the following expressions.

(a) \begin{align*}\sqrt[3]{\chi}\end{align*}χ3

(b) \begin{align*}\sqrt[4]{\chi^3}\end{align*}χ34


(a) \begin{align*}\chi^{\frac{1}{3}}\end{align*}χ13

(b) \begin{align*}\chi^{\frac{3}{4}}\end{align*}χ34

Evaluating Expressions with Exponents

It is important when evaluating expressions that you remember the Order of Operations. Evaluate what is inside the parentheses, then evaluate the exponents, then perform multiplication/division from left to right, and then perform addition/subtraction from left to right.

Example C

Evaluate the following expression.

\begin{align*}3 \cdot 5^2 - 9^{1/2} \cdot 5+1\end{align*}35291/25+1


\begin{align*}3 \cdot 5^2-9^{1/2} \cdot 5+1=3 \cdot 25-\sqrt{9} \cdot 5+1=75-3\cdot 50+1=75-150+1=-74\end{align*}35291/25+1=32595+1=75350+1=75150+1=74


Roots as Fractional Exponents: \begin{align*}\sqrt[m]{a^n}=a^{\frac{n}{m}}\end{align*}anm=anm.

Guided Practice

Simplify \begin{align*}\left(\frac{3^2\cdot 36^{3/2}}{2^3}\right)^{2/5}\end{align*}(32363/223)2/5.


We start by simplifying using the fact that \begin{align*}36^{3/2}=(36^{1/2})^3\end{align*}363/2=(361/2)3.

\begin{align*}\left(\frac{3^2\cdot 36^{3/2}}{2^3}\right)^{2/5}= \left(\frac{3^2\cdot (36^{1/2})^3}{2^3}\right)^{2/5}= \left(\frac{3^2\cdot (\sqrt{36})^3}{2^3}\right)^{2/5} = \left(\frac{3^2\cdot 6^3}{2^3}\right)^{2/5}\end{align*}(32363/223)2/5=(32(361/2)323)2/5=(32(36)323)2/5=(326323)2/5

Next we rearrange knowing that 6 and 2 have a common factor, 2.

\begin{align*}\left(\frac{3^2\cdot 6^3}{2^3}\right)^{2/5}=\left( 3^2 \cdot \left(\frac{6}{2}\right)^3\right)^{2/5} = \left( 3^2 \cdot 3^3\right)^{2/5} =\left(3^5\right)^{2/5} =3^{5\cdot 2/5}=3^2=9\end{align*}(326323)2/5=(32(62)3)2/5=(3233)2/5=(35)2/5=352/5=32=9


Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Zero, Negative, and Fractional Exponents (14:04)

Simplify the following expressions. Be sure the final answer includes only positive exponents.

  1. \begin{align*}\left(a^{\frac{1}{3}}\right)^2\end{align*}(a13)2
  2. \begin{align*}\frac{a^{\frac{5}{2}}}{a^{\frac{1}{2}}}\end{align*}a52a12
  3. \begin{align*}\left(\frac{x^2}{y^3}\right)^{\frac{1}{3}}\end{align*}(x2y3)13
  4. \begin{align*}\frac{x^{-3}y^{-5}}{z^{-7}}\end{align*}x3y5z7
  5. \begin{align*}(x^{\frac{1}{2}} y^{-\frac{2}{3}})(x^2 y^{\frac{1}{3}})\end{align*}(x12y23)(x2y13)

Simplify the following expressions without any fractions in the answer.

  1. \begin{align*}\left(\frac{3x}{y^{\frac{1}{3}}}\right)^3\end{align*}(3xy13)3
  2. \begin{align*}\frac{4a^2b^3}{2a^5b}\end{align*}4a2b32a5b
  3. \begin{align*}\left(\frac{x}{3y^2}\right)^3 \cdot \frac{x^2y}{4}\end{align*}(x3y2)3x2y4
  4. \begin{align*}\left(\frac{ab^{-2}}{b^3}\right)^2\end{align*}(ab2b3)2
  5. \begin{align*}\frac{x^{-3}y^2}{x^2y^{-2}}\end{align*}x3y2x2y2
  6. \begin{align*}\frac{3x^2y^{\frac{3}{2}}}{xy^{\frac{1}{2}}}\end{align*}3x2y32xy12
  7. \begin{align*}\frac{(3x^3)(4x^4)}{(2y)^2}\end{align*}(3x3)(4x4)(2y)2
  8. \begin{align*}\frac{a^{-2}b^{-3}}{c^{-1}}\end{align*}a2b3c1
  9. \begin{align*}\frac{x^{\frac{1}{2}}y^{\frac{5}{2}}}{x^{\frac{3}{2}}y^{\frac{3}{2}}}\end{align*}x12y52x32y32

Evaluate the following expressions to a single number.

  1. \begin{align*}(16^{\frac{1}{2}})^3\end{align*}(1612)3
  2. \begin{align*}5^0\end{align*}50
  3. \begin{align*}7^2\end{align*}72
  4. \begin{align*}\left(\frac{2}{3}\right)^3\end{align*}(23)3
  5. \begin{align*}3^{-3}\end{align*}33
  6. \begin{align*}16^{\frac{1}{2}}\end{align*}1612
  7. \begin{align*}8^{\frac{-1}{3}}\end{align*}813

Mixed Review

  1. A quiz has ten questions: 7 true/false and 3 multiple choice. The multiple choice questions each have four options. How many ways can the test be answered?
  2. Simplify \begin{align*}3a^4 b^4 \cdot a^{-3} b^{-4}\end{align*}3a4b4a3b4.
  3. Simplify \begin{align*}(x^4 y^2 \cdot xy^0)^5\end{align*}(x4y2xy0)5.
  4. Simplify \begin{align*}\frac{v^2}{-vu^{-2} \cdot u^{-1} v^4}\end{align*}v2vu2u1v4.
  5. Solve for \begin{align*}n: -6(4n+3)=n+32\end{align*}n:6(4n+3)=n+32.

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roots as fractional exponents \sqrt[m]{a^n}=a^{\frac{n}{m}}.

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